Spiritual and religous inheritance are two different things. If you are refering to spiritual inheritance you may be referring to Acts 26:17-18
The Lord said unto Paul, "I will send thee unto the Gentiles, To open their eyes, and to turn them from darkness to light, and from the power of Satan unto God, that they may received forgiveness of sins, and inheritance among them which are sanctified by faith that is in me".
In this case, inheritance refers to the Christian belief that Jesus the Christ, through his sacrifice, granted salvation to all Christians meaning that they are all promised the kingdom of heaven if they are believers.
If you are referring to our religious upbringing, "religious inheritance is somewhat of an ambiguous term but, judging by the rest of your question, I assume you refer to culture and the religious values and beliefs that are passed down from generation to generation and from culture to culture.
An example of this type of inheritance, particularly for people who live in the United States and are not part of a majority is Catholicism.
I will give you my own example: I am Hispanic, born and raised Roman Catholic, and still practicing for major things. For example, when there is a death or a terminal illness I pray the rosary. When there is a funeral, I truly believe that the individual's soul goes to purgatory and that their entrance to heaven will be aided by prayer. Whether that makes rational sense to me is a different story; however, the strength of such belief has been bestowed upon me from a very young age, and it has also been nurtured through life. For this reason, the effect of religious inheritance is that, when it is well-nurtured and consistently applied, it can last a lifetime. Moreover, it could serve as a support and comfort mechanism to keep hope alive, even when things come spiraling down.
As part of this effect comes another issue: when you are part of a minority religion, you are less likely to find places to worship if you move to another country, such as the USA. In my case, the Army moved me to the deep U.S. South, where almost everyone in the population is pure-bred Southern Baptist. The number of Catholics in my particular area is so reduced that Catholics look more like a sect than a religious group as hard as it is to establish a church. Yet, it is something that, working together, can be and will be accomplished. Yet, to older, less adapted people, lacking a place to worship can be a big deal because they may incur in feelings of guilt, or even moral fear. This is a situation that not everybody thinks about, but can certainly occur as a result of religious inheritance when you move to the USA and you are a minority.
What can be done to solve this situation?
What many people do to nurture their religious inheritance is to focus on the traditions that the religion brings with it. For example, fasting during lent, celebrating the Virgin Mary on her day, reuniting to do the rosary, attending Ash Wednesday celebrations (in some military bases without catholic churches priests are invited to perform the service), keeping images of patron saints, saying prayers at home. All of those traditional actions constitute the religious inheritance that build the values upon which we build our personality and our system of beliefs.